Propagation-induced revival of entanglement in the angle-OAM bases

Although the continuous-variable position-momentum entanglement of photon pairs produced by parametric down-conversion has applicability in several quantum information applications, it is not suitable for applications involving long-distance propagation. This is because entanglement in the position-momentum bases, as seen through Einstein-Podolsky-Rosen (EPR)–correlation measurements, decays very rapidly with photons propagating away from the source. In contrast, in this article, we show that in the continuous-variable bases of angle–orbital angular momentum (OAM), the entanglement, as seen through EPR-correlation measurements, exhibits a remarkably different behavior. As with the position-momentum bases, initially, the entanglement in the angle-OAM bases also decays with propagation, and after a few centimeters of propagation, there is no angle-OAM entanglement left. However, as the photons continue to travel further away from the source, the entanglement in the angle-OAM bases revives. We theoretically and experimentally demonstrate this behavior and show that angle-OAM entanglement revives even in the presence of strong turbulence.

1 Calculation of conditional position and angle uncertainties 1.1 Derivation of the formulas For a Gaussian pump with beam waist at the crystal plane z = 0, the two-photon wavefunction in the position basis at the crystal plane z = 0 is given by (15,43,44): where ρ s ≡ (x s , y s ) and ρ i ≡ (x i , y i ) are the transverse positions of the signal and idler photons, respectively at z, k = π/λ p , |ρ s | = ρ s , and |ρ i | = ρ i . Also, w 0 is the pump beam waist at z = 0, σ 0 = 0.455Lλ p /2π, L is the length of the crystal, and λ p is the wavelength of the pump field.
In addition to the position-momentum bases the down-converted photons are rendered entangled in the angle-OAM bases as well. We obtain the two-photon angle probability distribution by writing P (ρ s , ρ i ; z) of Eq. (2) in the polar coordinates using the transformations ρ s = (r s cos θ s , r s sin θ s ) and ρ i = (r i cos θ i , r i sin θ i ), where (r s , θ s ) and (r i , θ i ) are the polar coordinates of the signal and idler photons at z, etc. We therefore get: |ρ s + ρ i | 2 = r 2 s + r 2 i + 2r s r i cos(θ s − θ i ), |ρ s − ρ i | 2 = r 2 s + r 2 i − 2r s r i cos(θ s − θ i ), P (r s , θ s , r i , θ i ; z) = |A| 2 exp − r 2 s + r 2 i + 2r s r i cos(θ s − θ i ) 2w(z) 2 × exp − r 2 s + r 2 i − 2r s r i cos(θ s − θ i ) 2σ(z) 2 .
We then integrate P (r s , θ s , r i , θ i ; z) over the radial coordinates in order to obtain the two-photon angle probability distribution function P (θ s , θ i ; z) as: As calculated using Eqs.
(2) and (4), the conditional position and angle uncertainties behave differently upon propagation. Although it is very difficult to derive the general analytical expressions for the conditional position and angle uncertainties, we have obtained analytical expressions for how the conditional position and angle uncertainties scale with z in the near-and far-field regions.
1.2 Near-and far-field behaviours of the conditional position uncertainty The two-photon position probability distribution function is given by Eq.
(2). By setting ρ i = 0, we write the conditional positional probability distribution function P (ρ s |ρ i ; z) as and |ρ s | 2 = ρ 2 s . From Eq. (5), we obtain the conditional position uncertainty in the y-direction as ∆(y s |y i ; z) = 1 For the experimental parameters of interest, we have w 0 = 507 µm and σ 0 = 11.3 µm. Therefore, in the near-field region, we have w(z) ≫ σ(z) and thus the conditional uncertainty in the y-direction becomes In the far-field, we have w(z) ≪ σ(z) and thus the conditional uncertainty in the y-direction becomes ∆(y s |y i ; z) ≈ w(z) = w 0 1 + z 2 k 2 w 4 0 . (8)

Near-and far-field behaviours of the conditional angle uncertainty
The two-photon angle probability distribution function is given by Eq. (4). The conditional angle probability distribution function P (θ s |θ i ; z) is obtained by setting θ i = 0 in Eq. (4). As we are interested only in obtaining the near-and far-field scalings of the conditional angle uncertainty, we take P (r s , θ s , r i , θ i ; z) = P (r s , θ s , θ i ; z)δ(r s − r i ). Thus we write Eq. (4) as P (θ s , θ i ; z) = P (r, θ s , θ i ; z)r 2 dr.
Using Eq. (3) and the Mathematica software, we evaluate the above integral and obtain where P 0 = |A| 2 π/2 8 , The ratio of C and D can be written as In our experiments, we have w 0 = 507 µm and σ 0 = 11 µm. Thus we have w 0 ≫ σ 0 , and under this approximation we write the above ratio as Next, we study the behaviour of P (θ s , θ i ; z) in the near field regions. We make use of the fact that for θ i = 0, P (θ s , θ i ; z) is maximum at θ s = 0. Therefore, we have We next find the value of θ s at which P (θ s , θ i ; z) = P max (θ s , θ i = 0; z)/2, in which case θ s can be taken as the half-width of the conditional angle probability distribution function. We thus equate P (θ s , θ i ; z) = P max (θ s , θ i = 0; z)/2 or, Solving the above equation, we get two solutions for θ s : The angle uncertainty ∆(θ s |θ i ; z) can therefore be written as Using the approximation cos −1 x = 2(1 − x) for x ∈ [0, 1], we write the above uncertainty as: Substituting for C/D from Eq. (12), we obtain In the near-field regions, we have k 2 σ 2 0 w 2 0 ≫ z 2 . Therefore, we can write the angle uncertainty in the near-field regions as Thus in the near-field regions the angle uncertainty increases linearly with z. In the far-field, we use the fact that for θ i = 0, P (θ s , θ i ; z) is maximum at θ s = π. Therefore, in the far-filed we have P max (θ s , θ i = 0; z) = P 0 /[C − D] 3/2 . Now, proceeding in the similar manner as above, and using the far-field approximation z 2 ≫ k 2 σ 2 0 w 2 0 , we find the angle uncertainty in the far-field regions to be We thus find t hat i n t he f ar-field re gion th e an gle un certainty ∆(θ s |θ i ; z) be comes inversely proportional to z and as a consequence decreases upon propagation. Figure

Calculation of signal photon position uncertainty
From the two-photon probability distribution in Eq.
(2), we find the position probability distribution P (ρ s ; z) of the signal photon in the following manner where w s (z) = ∆(y s |y i ; z)/ 1 − (D/C) 2 is the position uncertainty of the signal photon. We now plot w s (z) as a function of z in Fig We find that in the near-and far-field w s (z) behaves as w s (z) ≈ w(z)/2 and w s (z) ≈ σ(z)/2, respectively.

Calculation of conditional momentum and OAM uncertainties
Using the two-photon wave-function in the position basis ψ(ρ s , ρ i ; z) calculated in the previous section, we calculate the two-photon wave-function ψ(p s , p i ; z) in the transverse momentum basis, which is given by (44) where p s ≡ (p sx , p sy ) and p i ≡ (p ix , p iy ) are the transverse momenta of the signal and idler photons, respectively. Using the above equation, we find that the conditional momentum probability distribution function P (p s |p i ; z) of the signal photon is given by where p 2 s = |p s | 2 . The standard deviation of P (p s |p i ; z) in the y-direction ∆(p sy |p iy ; z) is the conditional momentum uncertainty of the signal photon. The above equation shows that P (p s |p i ; z) is independent of z and that ∆(p sy |p iy ; z) does not change upon propagation.
For a Gaussian pump the two-photon state produced by SPDC in the OAM basis can be written as (17) where l s ℏ and −l s ℏ are the OAMs of signal and idler photons, respectively. The form of the two-photon state above implies that if the signal photon is detected with OAM l s ℏ, then the idler photon is guaranteed to be detected with OAM −l s ℏ. For the above state, and with l i = 0, the conditional two-photon OAM probability distribution function takes the following form: P (l s |l i ; z) = S ls δ ls,0 . This implies that the corresponding conditional OAM uncertainty ∆(l s |l i ; z) is equal to zero. However, in an experimental situation, one always measures ∆(l s |l i ; z) to be non-zero (17). There are several reasons for this, which includes the pump not being an ideal Gaussian beam, the experimental imperfections such as misalignment and background noise, and the mode dependent detection efficiencies of OAM d etectors. These cause an additional contribution in P (l s |l i ; z) measurement. Therefore, in our experiments, we model the conditional OAM probability distribution function as: where S 0 , σ f and N are the fitting parameters. We take the width of P (l s |l i ; z) as the conditional OAM uncertainty ∆(l s |l i ; z).

Coincidence Measurement with EMCCD camera
In this section we outline how we use an Andor iXon Ultra-897 EMCCD camera having 512 × 512 pixel grid with 16 × 16 µm 2 pixel-size for measuring coincidence counts in position and angle bases. For this, we record 10 6 -10 7 images of the SPDC field with an exposure time of 1 ms -5 ms over a few hours with average flux of 0.5 -2.0 photons per pixel. We operate the camera at -60 • C with the electron-multiplication gain of 1000, the horizontal pixel readout rate of 5-17 MHz, the vertical pixel shift speed of 0.3 µs, and the vertical clock amplitude of +4V. In SPDC, a signal and idler photon pair gets generated within a very short time interval, usually of the order of 100 fs, which is much smaller than the exposure time (1 − 5 ms) of the EMCCD camera. Therefore, in all likelihood, the signal and idler photons belonging to a pair arrive within the same image. However, within the same image, we can also have signal and idler photons that are not from the same down-conversion pair. These give rise to the accidental coincidences, which between the pixels (or pixel groups) p and q can be estimated by computing the coincidence counts between k th and (k +1) th images. Therefore, as detailed in Ref.
(52,53), the true coincidence count C pq due to down-converted pairs between two pixels or two groups of pixels, p and q, of the EMCCD camera can be expressed as where the first term is the total coincidence and the second term is the accidental coincidence. The top and the bottom images represent the total coincidence and the accidental coincidence calculated using the first and the second terms of Eq. ( 29), r espectively. Subtraction of these two terms gives (c) the measured two-photon angle probability distribution function P (θ s , θ i ; z).
For measuring the two-photon position probability distribution function P (y s , y i ; z), we take millions of images using the EMCCD camera. For each image, we group the pixels into horizontal strips, y s and y i , as shown in Fig. S 4(a). The coincidence count between y s and y i can be written using Eq. (27) as where n For measuring the two-photon angle probability distribution P (θ s , θ i ; z), we group the pixels for each image into angular sectors as shown in Fig. S 5(a). The coincidence count between the angular sectors at θ s and θ i is given by: where n (k) θs and n (k) θ i are the photon counts of angular sectors θ s and θ i respectively. The top image in Fig. S 5(b) represents the total coincidence as a function of θ s and θ i , and it is evaluated using the first t erm o f E q. ( 29). T he b ottom i mage i n F ig. S 5 (b) r epresents t he accidental coincidence as a function of θ s and θ i , and it is evaluated using the second term of Eq. (29). The difference of these two images is proportional to the two-photon angle probability distribution P (θ s , θ i ; z), as shown in Fig. S 5(c). At θ s = θ i , the correlation becomes artificially perfect because we are correlating a pixel with itself. So, we discard the points with θ s = θ i as outliers.  P (y s ,y i ;z) P (y s ,y i ;z) P (y s ,y i ;z) P (y s ,y i ;z) P (y s ,y i ;z) P (y s ,y i ;z) ing the transverse planes between z = 10 and z = 60 cm, we keep the magnification to be 0.25. We scale the measured P (y s , y i ; z) such that its maximum value is equal to one.
We find that the photons are correlated in position in the near field whereas they get position anti-correlated in the far-field. In order to extract ∆(y s |y i ; z) from the measured P (y s , y i ; z), we fit P (y s , y i ; z) with the function: P f (y s , y i ; z) = bP r (y s , y i ; z) + aP n (y s , y i ; z), where ] is considered as the probability distribution due to the down-converted photons, while P n (y s , y is considered as the noise contribution. Here b, a, σ 1 (z), σ 2 (z), d, f , m and n are the fitting parameters. We consider n ≫ σ 1 (z), m ≫ σ 2 (z) such that the noise contribution remains much broader than the twophoton position probability distribution. The width ∆(y s |y i = 0; z) can now be expressed as ∆(y s |y i = 0; z) = σ 1 (z)σ 2 (z)/ σ 2 1 (z) + σ 2 2 (z). Figure S 6(c) shows ∆(y s |y i = 0; z) as a function of z. The simulation has been carried out using the expression given in Eq.
(2). Figure S 6(b) shows the experimentally measured P (θ s , θ i ; z) at different z. We scale P (θ s , θ i ; z) such that its maximum value is equal to one. The P (θ s , θ i ; z) plots show that near the crystal plane the signal and idler photons have the highest probability of arriving at the same angular positions. However in the far-field t he s ignal a nd i dler p hotons have the highest probability of arriving at angular positions separated by π radians. We fit the measured P (θ s , θ i ; z) with the analytic function: P f (θ s , θ i ; z) = bP r (θ s , θ i ; z) + a, where P r (θ s , θ i ; z) = 1/(1 + q cos(θ s − θ i − c)) 3/2 . Here, b, a, q, and c, are the fitting parameters. We derive the fitting function by putting r s = r i in Eq. ( 4). Next, we evaluate ∆(θ s |θ i ; z) by finding the standard deviation of P r (θ s |θ i ; z ) at various z v alues. Figure

Measurement of the two-photon momentum and OAM probability distribution functions
Equation (24) shows that P (p s |p i ; z) is independent of z and that ∆(p sy |p iy ; z) does not change upon propagation. For the given experimental parameters, the calculated value of ∆(p sy |p iy ; z) is 1.97ℏ mm −1 . The two-photon OAM probability distribution function P (l s , l i ; z) remains unchanged as a function of z. We verify this by making several measurements of P (l s , l i ; z) as a function of z. We plot the experimentally measured two-photon OAM probability distribution function P (l s , l i ; z) and the conditional OAM probability distribution function P (l s |l i ; z) at z = 50 cm in Figs. S 7(a) and S 7(b), respectively. As described in Sec. 3, we fit the conditional distribution with the analytical function P (l s |l i ; z) = S ls δ ls,0 + N exp −l 2 s /(2σ 2 f ) , where S 0 , N and σ f are the fitting parameters, and thus find the uncertainty ∆(l s |l i ; z) to be 0.72ℏ radian −1 in our experiments.

Measurement of the two-photon angle probability distribution function in turbulence
In this section, we present how the conditional angle uncertainty propagates after the twophoton field passes through turbulence. Figure S 8(a) illustrates the propagation of SPDC photons through a planar turbulence kept at a distance z = d from the crystal plane located at z = 0. We are interested in finding the two-photon angle probability distribution function at a propagation distance z. The presence of turbulence introduces statistical randomness in the two-photon field, and so we need to describe the field propagation in terms of the two-photon cross-spectral density function. From z = 0 up to z = d, the two photon field remains pure and can be described by the two-photon wave-function ψ(ρ s , ρ i ; z). Therefore, the two-photon can be written as where, ρ ′ s ≡ (x ′ s , y ′ s ), and ρ ′ i ≡ (x ′ i , y ′ i ) are the transverse co-ordinates of signal and idler photons respectively, at z = d.
is the two-photon cross spectral density function right before the turbulence plane. The effect due to the turbulence is captured through the cross-spectral density function W turb (ρ ′ s1 , ρ ′ s2 , ρ ′ i1 , ρ ′ i2 ), which we approximate by modelling the turbulence in terms of a Gaussian function where r is the turbulence strength (55). We rewrite the above equation as where ψ(ρ ′′ s , ρ ′′ i ; 0) is the two-photon wave-function at the crystal plane z = 0 and is given by we find the two-photon cross-spectral density function at z and thereby the two-photon position probability distribution function P (ρ s , ρ i ; z): By substituting Eq. (32) into Eq. (33), we compute P (ρ s , ρ i ; z) as a function of z. We then use the transformations ρ s = (r s cos θ s , r s sin θ s ) and ρ i = (r i cos θ i , r i sin θ i ) in order to obtain P (θ s , θ i ; z) using Eq. (33). Figure S 8(c) shows the experimentally measured P (θ s , θ i ; z) at different z (see Sec. 5 for the measurement details). Figure S 8 (b) shows the P (θ s , θ i ; z) calculated using Eq. (4) at different z for the relevant experimental parameters of d = 15 cm, L = 5 mm, w 0 = 507 µm. For the simulation plots, we use the turbulence strength r as a fitting parameter and find its value to be 0.125 mm. We note that the experimentally measured P (θ s , θ i ; z) contains some noise distribution, which gets prominent at large z. This can be attributed to the fact that in the experiment, we insert a distributed turbulence in the path of the two-photon field, whereas in the simulations, we approximate that as a planar turbulence. Nevertheless, the diagonal correlation in the experimentally observed P (θ s , θ i ; z) matches with that of the numerical simulations. Using the procedure described in Sec. 5 B, we extract the conditional angle uncertainty ∆(θ s |θ i ; z) from Figure  8 Measurement of the two-photon OAM probability distribution in turbulence In this section, we present a theoretical model to evaluate the influence of turbulence on the conditional OAM distribution P (l s |l i ; z). Within paraxial approximation and the Gaussian pump beam assumption (17), the OAM remains conserved in SPDC. This means that if the idler photon is detected with OAM l i ℏ = 0, the signal photon is guaranteed to be detected with OAM l s ℏ = 0. For the measurement of conditional OAM distribution we choose l i = 0.
9 Effects of a finite-size detector on angle-OAM entanglement In our analysis so far, we have considered the photon detectors to have an aperture of infinite extent, as we take the upper limit of radial integration to be ∞ in Eq. (4). However, experiments are always carried out with detectors of finite extent. In this section, we numerically analyze the effects of using finite-size detectors on the revival of entanglement in the angle-OAM entanglement bases. The two-photon angle probability distribution function P (θ s , θ i ; z) in this case is given by P (θ s , θ i ; z) = D/2 0 D/2 0 r s r i P (r s , θ s , r i , θ i ; z)dr s dr i , where D represents the diameter of the detector. Using the above equation, we numerically calculate ∆(θ s |θ i ; z) as a function of z upto a propagation distance of z = 1 km for different values of D. We use the conditional OAM uncertainty ∆(l s |l i ; z) = 0.72ℏ corresponding to to 1 km, we plot ∆(θ s |θ i ; z)∆(l s |l i ; z) as a function of ln(z) using the relevant experimental parameters. We see that for all values of D, the entanglement revives at z = 24 cm. We find that due to finite D, the product ∆(θ s |θ i ; z)∆(l s |l i ; z) starts increasing upon long-distance propagation and beyond a certain z the entanglement is not witnessed in the angle-OAM bases.
Furthermore, we find that the increase in diameter D increases the propagation distance upto which the entanglement in the angle-OAM bases remains intact. Therefore, one can decide the minimum aperture size for a given application based on the distance upto which one needs the entanglement in the angle-OAM bases to be intact.